Question

In a class, 5% of the boys and 10% of the girls have an IQ of more than 150. In this class, 60% of the students are boys. If a student is selected at random and found to have an IQ of more than 150, find the probability that the student is a boy.

Answer 1

Let A, E₁ and E₂ denote the events that the IQ is more than 150, the selected student is a boy and the selected student is a girl, respectively.

$$\begin{gathered} \therefore P\left(E_1\right)=\frac{60}{100} \\ P\left(E_2\right)=\frac{40}{100} \\ \mathrm{Now}, \\ P\left(A/E_1\right)=\frac{5}{100} \\ P\left(A/E_2\right)=\frac{10}{100} \\ \mathrm{Using}\mathrm{Bayes}\text{'}\mathrm{theorem},\mathrm{we}\mathrm{get} \\ \mathrm{Required}\mathrm{probability}=P\left(E_1/A\right)=\frac{P\left(E_1\right)P\left(A/E_1\right)}{P\left(E_1\right)P\left(A/E_1\right)+P\left(E_2\right)P\left(A/E_2\right)} \\ =\frac{{\displaystyle \frac{60}{100}}\times {\displaystyle \frac{5}{100}}}{{\displaystyle \frac{60}{100}}\times {\displaystyle \frac{5}{100}}+{\displaystyle \frac{40}{100}}\times {\displaystyle \frac{10}{100}}} \\ =\frac{300}{300+400}=\frac{300}{{\displaystyle 700}}=\frac{3}{7} \end{gathered}$$